Let (In) be a nested sequence of closed bounded intervals. For each natural number n, let xn be in In. Use the Bolzano-Weierstrass Theorem to give a proof of the Nested Intervals property.

Intervals is ? Do you have to use BW? It's easier to use the fact that since each interval is bounded and the diameter of the sets approaches zero (which I assume is what you want) then we get by the completeness of [tex]\mathbb{R}[/mat] that the intersesection contains a single point. Otherwise, argue that if we have that each must converge to a point by BW. And use the diameter approaching zero to show that it the two points must be equal.